Question: $ 0.\overline{70} \div 2.\overline{43} = {?} $
First convert the repeating decimals to fractions. $\begin{align*} 100x &= 70.707...\\ x &= 0.707...\end{align*} $ $\begin{align*} 99x &= 70 \\ x &= \dfrac{70}{99}\end{align*} $ $\begin{align*} 100y &= 243.4343...\\ y &= 2.4343...\end{align*} $ $\begin{align*} 99y &= 241 \\ y &= \dfrac{241}{99}\end{align*} $ So, the problem becomes: $ \dfrac{70}{99} \div \dfrac{241}{99} = {?} $ Dividing by a fraction is the same as multiply by the reciprocal of that fraction. $ \dfrac{70}{99} \times \dfrac{99}{241} = {?} $ $ \phantom{\dfrac{70}{99} \times \dfrac{241}{99}} = \dfrac{70 \times 99}{99 \times 241} $ $ \phantom{\dfrac{70}{99} \times \dfrac{241}{99}} = \dfrac{70 \times \cancel{99}} {\cancel{99} \times 241} $ $ \phantom{\dfrac{70}{99} \times \dfrac{241}{99}} = \dfrac{70}{241} $